We use to consider space as a continuum, in which the physical phenomena happen, particles and fields.
A common analogy is with the surface of a fair balloon, a kind of rubber membrane, on which the phenomena, objects, etc. are painted. Our universe is (as far as we know), an hypersphere, this meaning the three-dimensional equivalent of the surface of a sphere (which has two dimensions). And, just like the phenomena painted on the surface of the sphere can not leave this surface, are forced to stick to it, so the physical phenomena of our universe are glued to, or contained in, the equivalent of the rubber membrane, but in three dimensions. Useless to try to figure these bizarre entities, our mind is not able of this. What we just need to remember is that the objects of our experience are like «painted» on the equivalent of a surface, but in three dimensions, what the physicists call a «brane» (generalization of a membrane, but with more than two dimensions). This is called the space-time continuum, and a common image to view the gravitational field of a planet is a ball which distorts the three dimensional «membrane», like a ball posed on a two dimensional rubber membrane.
This picture perfectly describes large-scale phenomena, space, cosmology, gravitational fields of the planets. But does it match a reality? Is it there really a «membrane», and if so, what is its nature? Quantum mechanics explains very well how particles and fields behave, without requiring anything else. The theory of the logical self-generation process clearly explains quantum mechanics, without requiring anything else. We can certainly assume that the membrane would be «something» which would maintain the phenomena in the three-dimensional space, while obeying Relativity. But this makes that this «membrane» would have very complex properties! As a super-aether, but which, this time, would be relativistic.
However, we saw in chapter III-4 that the process of logical self-generation is perfectly able of generating the equivalent of a three-dimensional space, quite similar to ours, and indistinguishable from ours, for the people who would live into such a process. This offers the possibility to get rid of the «membrane» (the space-time continuum existing independently of its content).
So we will pose that the structure of the three-dimensional space that we observe in our universe is simply, to speak the language of the Sets Theory, the structure of the set of the nibs, just as sets of numbers, such as R3 or the set of the trinomials, also have a structure of a three-dimensional space, totally indistinguishable of the one of our universe.
And, useless to add, the sets need no «rubber membrane» to keep the trinomials perfectly aligned in their three dimensions. The physical universe can well do the same.
However, the example of a logical self-generation process seen in chapter III-4 had nibs of space and nibs of particles. However our universe has only nibs of particles. Then, how space is generated?
Let us see an example: a scientist sends a photon on one of the reflectors installed on the Moon, and gets it back. Note that this photon undergoes no quantum interaction on this path, the mirror only changing its point of reappearance. What is interesting, is that the photon arrives precisely where we expect it. And it seems that this accuracy can be improved almost without limit: if the errors of measurement or calculation approach zero, then the trajectory approaches perfection.
But there is better. Indeed, in a reflection on the nature of space, we cannot pose a priori that the photon will find back our three dimensions space. Indeed, if there is no absolute space, then the photon, or any other particle, can also move away in a fourth, a fifth dimension... and never see our world again. The fact is that it finds it back: it reaches the intended target, not only where we expect it in our three dimensions space, but also, right into this three dimensions space! And this time with an absolute accuracy, without any possible error or disturbance.
So everything happens as if we had a «membrane», an absolute space guiding the particles in an absolute way, and forcing them to stay in this space. However, the theory of the logical self-generation does not provide for such an object, which requires other ad-hoc causes. We must therefore admit that this accuracy is not caused by a continuum guiding the particles, like the surface of the balloon which guides the drawings printed on it. This accuracy would come, once again, from the perfect repeatability of physical phenomena. Just like time, which seems absolute and indefinitely repeatable, while not intrinsically defined, then the trajectories of particles make them stay precisely in our familiar three dimensions, with nothing to force them to do so, with a perfect accuracy that nothing can disturb. This perfect repeatability of physical phenomena is required by the Rule 3 (Case 5) of chapter III-3: the successive inferences of the logical self-generation system are logically forced to always reproduce according to perfectly identical laws.
So we did not really have a space which would be knitted stitch per stitch, as in chapitre III-4, but particles that are held perfectly in it, with nothing to guide them: the vacuum is truly empty, immaterial, without proper existence, without structures or properties. And there is no need for «nibs of space» between the material particles, and not either a medium between particles to transmit the forces.
So our three-dimensional vacuum is just the place where our mathematical particles always meet, compared with a larger vacuum with four dimensions or more. Our universe is an incredible scaffolding, where huge superstructures meet perfectly, ad infinitum, with the absolute accuracy of their elements, like an assembly of perfect bricks: if a first brick is positioned, then it determines a mesh where all the other bricks are forced to take place, ad infinitum. With these conditions, there is really nothing needed to guide this scaffolding.
To get an ideas, a good example would be a formation of meteorites flying together, forming a plan. The various forces affecting them may tilt the plan, but the meteorites will continue to keep this figure, even if they move apart, collide, and so on. So they really generate a surface, from the only play of their inertia, without the need for anything to guide them, or hold them together. Our three-dimensional space would also be generated in this way, with all its particles flying together, while keeping the configuration of the three-dimensional space. The only difference is that the inaccuracies of the movement of the meteorites would quickly break the plan, while the particles of our universe always stay in the three-dimensional space with a perfect precision, never breaking up, even after billions of years. And regardless of their movement within these three dimensions. For instance, two meteorites bumping in each other are likely to leave their plan, but our physical laws make that, even when they collide, our particles do not leave our three-dimensional space.
Another way to say the same thing is to use again the comparison with the sets of trinomials. Let us suppose a logical self-generation process, a series, creating new trinomials from others. Even if we introduce uncertainties into the calculus, and thus cannot predict exactly where are the next trinomials, it is however obvious that, whatever their value, these new trinomials will still be exactly in the three dimensions of their trinomial space. It is just mathematically impossible to be otherwise, as trinomials have, by definition, three dimensions. If our universe is in the same situation, all its elements are mandatorily into its three dimensions, and cannot be anywhere else. With no need of anything, concrete or abstract, to keep these elements in these three dimensions.
Where we are now, we can ask the question, why three dimensions? Why not two, why not four?
The simplest answer is precisely that there is no reason, and that this would be one of the arbitrary and intrinsic properties of the nibs of our universe, as to organize themselves in three dimensions, as seen in chapter III-3. Other universes may arbitrarily have two or four dimensions, or even more complex structures.
However, other more complicated answers are possible, which are about the causes of the laws of physics. For this reason, we shall study them only after chapter IV-6 on anthropism.
However the reality is more complex. Indeed, quantum mechanics says that the fields in the vacuum (electric field, magnetic field, etc.) have a complex structure, quantified. This results into virtual particles, appearing and disappearing into the vacuum. And this even when these fields are all null. This is what is sometimes called vacuum energy.
The virtual particles are the same as the «real» particles of which our bodies are made. However, having no energy, they have no mass, and therefore no direct influence on the «real» particles. In addition, they disappear immediately after being created. This is why the vacuum seems really empty, and not as a gas. But these virtual particles still have an influence on the real particles, not by producing observable quantum interactions, but by influencing the interactions which occur between the real particles. This structure of vacuum would be the cause of various bizarre physical effects, such as the Casimir effect (an attractive force between two close conductors, verified) the Unruh effect (photons appearing in an accelerated reference frame, hypothetical) or the Hawking effect (The radiation of black holes, hypothetical).
This structure of vacuum also plays an important role in the properties of visible particles, for instance by adding mass to them. So, a proton or a neutron are made up of three quarks, but the bulk of their mass comes from a large number of virtual particles, mesons and gluons, attracted by these quarks and inseparable from them. These particles also form a «screen», which makes the strong interactions of the quarks undetectable, as soon as we get at some distance of the proton or the neutron.
The theory of the logical self-generation process easily accommodates these virtual particles, if we consider each of them (from its appearance to its disappearance) as a single nib (or pair). Their relationship are then ordinary quantum relations, the same as those which occur between «real» particles. The only difference is that a real particle is formed of a succession of nibs in time. As mass cannot disappear, a nib with a mass necessarily produces another with the same mass. Thus a real particle is a chaining of nibs which transmit their mass and other charges, the one to the other. But in the case of virtual particles, there is no continuation to the chain, because there is no mass to pass: the particle disappears immediately after being created. Other properties, such as the electrical charge, cancel, because the law of conservation of charges makes that it is always formed two particles of opposite charge.
The same causes producing the same effects, these virtual particles, with the same properties than the «real» ones, will also organize themselves according to our familiar three-dimensional space. Many scientists then consider that these virtual particles, and their relationships, form a continuum which defines the physical space with three dimensions. In my opinion, with the theory of logical self-generation, this is not necessary, and the virtual particles do not play a more fundamental rôle than the «real» particles, about the definition of space. They only are in it. The only difference is that, unlike the «real» particles, which can be removed with a vacuum pump, we cannot remove the virtual particles out of space. If we take the image of chapter III-4, they occupy the place of the nib of space. But they are not even required to define the space, no more than the «rubber membrane».
So, if we could cut out a portion of space, and remove everything in it, «real» or virtual, nib or relationship, and then put our hand in this place, then we would notice nothing special, just ordinary vacuum, with exactly the same properties, the same dielectric constant, the same virtual particles. Electric field, gravitational field, etc.. would be transmitted in the same way as in ordinary vacuum. Whereas if there was a space-time continuum, a «rubber membrane» required to transmit the fields, then removing a piece of it would create a totally black form, with very different properties of the usual vacuum, and not transmitting fields at all, just as the Cavorite of HG Wells. And if we try to poke this thing, our finger would stop existing…
And why this «real» vacuum would have the same dielectric constant than the natural vacuum? By which fantastic coincidence? Simply because this dielectric constant is not a property of the vacuum, but of the matter which is there, that is, ultimately, of the logical self-generation process which is propagating here. The vacuum itself has no properties, even not to have three dimensions. The constants of vacuum that we measure are not a property of this vacuum, but of the matter of which the measuring instrument is made.
At a pinch, if we could put together into the same vacuum chamber, two different measurement apparatus belonging to two different self-generation processes (two «domains», with the wording of physics), then each would measure its own constant, into the same vacuum.
And when scientists of the Big Bang theory say that different kinds of vacuum («textures» or «domains») existed in different historical time of the universe (or in different places), this means, in our theory, that each of these domains is a different logical self-generation processes, which each create a space-time of its own, with different properties, different basic constants, and different laws of physics. But all of these propagate into the same nothingness, devoid of any properties or dimensions. And each transition from one to another occurred when a logical paradox forced a change in the self-generation law, as seen in chapter III-4, rule 6. So we see why I called this process «creative absurdity»... we owe it our very existence!
That the vacuum is truly empty explains at least one of its properties: not to hinder the movement. When the theory of vacuum was the aether, it had to exhibit very contradictory properties, such as to be extremely rigid to transmit the electric field, while being... perfectly fluid, to exercise no friction on moving bodies. This problem is not really solved with the idea of a «rubber membrane» continuum, even relativist. However, no such contradiction appears when we considers that space is only the structure of the set of the nibs, without any other form of reality. And a set structure never braked anything :-)
Last remarks, physicists of the Big Bang often consider a «quantum vacuum» (or «quantum state») of which emerged our universe. After the logical self-generation theory, this state would be a state where no laws of physics define reality, so that anything can happen, such as in chapter III-4, rule 3. And, in a general way, any physical state involving virtual particles or random fluctuation (Unruh effect, Hawking effect) would be simply situations where the current laws of physic do not fully define reality, allowing for the appearance of these random elements. So that, again, these elements would not appear out of a vacuum with mysterious structures and properties, but as elements of the ongoing logical self-generation process, existing in the space structure generated by this process.
We tend to think that each particle, each event, occurs in a perfectly defined place and time, that we could measure with an accuracy converging to perfection. This is not true: according to quantum mechanics, the position of the particle can not be defined absolutely. This is not because there would be errors impossible to eliminate, but these quantities are simply not defined in an absolute way. We saw the example of the electron around an atom, which position is a cloud of probability of presence extending over some area of space. This, the world of the particles is a fuzzy world, clouds brushing against each other, and which suddenly reify in different configurations, still as fuzzy.
The scale relativity, from the physicist Laurent Nottale, is another blow to the notion of absolutely accurate position: if we consider small enough structures, trying to observe them with a more magnifying «microscope» does not change the image obtained.
In a general way, physicists consider that, at a very small scale, space is no longer Galilean: it forms bumps, hollows, and even «drops» which stand out. Thus the particles are not necessarily strictly in our usual three dimensions, they more or less deviate of it. (They do it, however, only for very short durations. With time they always come back in our familiar flat three dimensions).
Under these conditions, with in more Special Relativity, which distorts our perception of events according to our speed, and General Relativity, which straightforwardly folds space according to the gravitational field, it is clear that our notion of absolute three-dimensional space (called Galilean) matches the physical reality only under certain conditions (the ones we experience in our everyday life on Earth). Beyond, the concept of absolute position, or continuous space, has no sense, no more than the absolute time of chapter IV-3. We can also compare it to the notion of «flat Earth», which satisfies only our senses.
So we must abandon these two concepts, and consider in a completely different way the way space is generated by the logical self-generation process, without any reference to an underlying absolute space, or even not to our familiar three dimensions.
We, since the beginning of this part, reasoned as if the nibs, or the particles, had a specific place, and figured their behaviour as moves in our familiar three-dimensional space. However, this way of thinking does not really allows to understand what the space is, and how it appears.
Even the scientists make this mistake. I remember seeing a huge computer simulation, with the supercomputer, of the structure of the nucleon, to calculate its mass. Well, great job. But the mass is, according to General Relativity, curved space. But if we place the elements of a simulation in the coordinates of a Galilean space, it is strictly impossible to curve this said space. Thus, the simulation only allowed to calculate the mass indirectly, without explaining how the nucleon curves the space around it.
It is then clear that we must first calculate the relations between the elements of the simulation.
Only after, we can try to see how they place themselves in space, without prejudice about the form or the number of dimensions of this space.
This would require to calculate in the reverse way than usual. In place of having variables as functions of x, y, z and t, we would in the contrary have the distances between each pair of particles, which would be functions of the quantum state of each particle.
I think it is theoretically possible, using the reverse functions of the usual ones. Being not proficient in the mathematical apparatus of quantum mechanics, I have no idea how to do it, and of any involved difficulties. I even not know if the calculations would be simplified or more complex. It may for example happen that the usual Galilean space is a kind of «epicycle» which would unnecessarily complicate the calculations (as formerly the epicycles of the Ptolemaic astronomy, trying to calculate the motions of the planets starting from the misconception as what the Earth would be at the centre). If this is true, then we might arrive at a simple presentation of quantum mechanics... It would be fascinating.
An argument in this way is that a quantum computation often requires integrating values on the whole space, or on a set of possibilities such as the Hamiltonian. It could happen that calculating without space replaces these integrals with simple integer values, reducing the quantum computing to an much more manageable arithmetic than the integrals or vector analysis. It is to add space which would complicate the calculations, for phenomena which, basically, are not located in space.
So we come to a completely different vision of the usual spatiocentrism: nibs which self-generate in direct logical relation the one with the other, in a pure Feynmann diagram, without any notion of space or time, with just causal logical relationship between each other.
Let us continue on the same image as before: The nibs are connected two by two by «sticks», representing their logical relationships. We can then assign lengths to these sticks, depending on the intensity of the logical relationship. So, if we have two nibs, they define a space in one dimension. Three nibs define a two dimensions space, four a three dimensions space, etc.. Thus N nibs define a space with N-1dimensions, and they each have a position in such a space.
If the number of nibs (and thus of particles) becomes important, then the notion of distance gets a meaning: it is likely that the nibs with intense interaction are «close», and those with weaker interactions are «far». What we should then observe, by calculating in this way, is that a large number of nibs, while still being in a space with many dimensions, organize themselves in a structure very similar to our relativistic space-time with three dimensions of space and one of time. And the more we increase the size of this set of nibs, the closer we are of our classical three-dimensional vision, to the point that living beings, at their level, perceive a perfect three-dimensional world.
Thus our three-dimensional space would be only the statistical average of the positions of the particles in a space which number of dimensions would be virtually infinite. We saw earlier that our three-dimensional space is the structure of all the nibs, in the meaning of the Sets Theory. This is still true, except that it is only a statistical average, just as with the pressure and air temperature, which look uniform, but which are just the statistical average of the irregular movement of the molecules .
And the difference between the average (the 3D space) and the exact position of each nib is given by this notion of space becoming «rough» at small scale, or which would be filled with «quantum fluctuations» (or the Heisenberg uncertainties, although it is somewhat different). In fact there is no rough space: there is simply no defined space at this scale, just as there is nothing between the squares of the chess game. Simply, each nib is closer or farther of the average, the perfect 3D space.
We already seen previously an «universe» of the chess game: In this universe, if scientists try to magnify their image larger than the size of the squares, then they will also observe a «scale relativity» quite similar to ours... simply because, at some point, «smaller» is meaningless.
In this way, we say that space is not a fundamental property of the universe (continuum, membrane ...) but an emergent property.
This way of seeing things was published in 2009 by the physicist Erik Verlinde. However I claim the paternity of it, as I was already describing it in 2000, in the version 1 of this book «General Epistemology», chapter 39, ISBN 0-75960-349-9, registered in the Library of Congress, Washington.
The idea would be to do a simulation of an entire universe, including a small number of particles. However, their relationship would be calculated nib by nib, without a priori placing the particles into a space. The properties of these particles would be what quantum mechanics predicts. Only after building the Feynman diagram of the whole set, and calculating all the interactions two by two, these n nibs would be placed in a space with n-1 dimensions. One difficulty here is that each of the «sticks» connecting the nibs two by two may actually have several different lengths, so a statistical approach will probably be needed. What I predict is that the application of statistical regression functions to all the nibs should allow to discern a three dimensions space, obeying the known laws of physics, with the known properties of space (gravitational constant, dielectric constant...).
If in more we use the relativistic nib seen in chapter IV-5, then this universe will obey the laws of special and general Relativity.
So, Relativity should be deduced from quantum mechanics, in this way, without any need to add something to it.
(It may however be possible to apply only a simple statistical regression, based on a Galilean space, corresponding to the relativistic concept of «local observer»)
However the calculation of emergent properties, starting from fundamental properties, is one of science's most complex and poorly understood domains. For instance, we can even not calculate the melting point of water from the properties of its molecule, despite it is the most studied. At the current rate of increase in power of computers, such simulations will not be possible before many years.
Today conceptions of physics offer nothing allowing to bypass this physics, and especially no mean to get rid of the speed of light. So, the hyperspace remains, for classical physicists, pure science fiction. Even not anticipation.
However it may happen that the three-dimensional space is not the only solution to describe the set of nibs. Under these conditions, we can speculate on the possibility of an hyperphysics, which would be only another way of describing the same physics, and the same everyday objects, but in a different emergent space, such as five-dimensional, or fractal. We can hope that such an hyperphysics would allow to do things impossible to the classical physics, such as teleporting objects, exchange of information, or travelling beyond the speed of light. What I think is the most likely would be a kind of machine where an apparent quantum random would reveal information about objects in the future, or away into space.
Ideas, texts, drawings and realization: Richard Trigaux.
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